Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

From Hodge theory for tame functions to Ehrhart theory for polytopes

Antoine Douai

Abstract

We study the interplay between Sabbah's mixed Hodge structure for regular functions and Ehrhart theory for polytopes. To this end, we analyze the properties of the Poincaré polynomial of the Hodge filtration of this mixed Hodge structure.

From Hodge theory for tame functions to Ehrhart theory for polytopes

Abstract

We study the interplay between Sabbah's mixed Hodge structure for regular functions and Ehrhart theory for polytopes. To this end, we analyze the properties of the Poincaré polynomial of the Hodge filtration of this mixed Hodge structure.

Paper Structure

This paper contains 17 sections, 31 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. Framework: Sabbah's mixed Hodge structures for regular functions
  3. Hodge-Ehrhart theory for tame regular functions
  4. $\theta$-vectors
  5. Hodge-Ehrhart polynomials
  6. Reflexive and anti-reflexive functions
  7. Coefficients and roots of Hodge-Ehrhart polynomials
  8. Thom-Sebastiani and Hodge-Ehrhart polynomials
  9. Sabbah's mixed Hodge structures and Ehrhart theory
  10. Hodge and Ehrhart via Newton polytopes
  11. Relations with prior results in combinatorics
  12. From singularity theory to combinatorics
  13. From combinatorics to singularity theory: the Lower Bound Theorem
  14. Linear inequalities among the Hodge numbers of Sabbah's mixed Hodge structures
  15. The $H$-conjecture
  16. ...and 2 more sections

Key Result

Lemma 2.2

The multiplication by $\tau^{-1}$ induces isomorphisms if $\alpha\in ]0,1[$ and

Theorems & Definitions (58)

  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Remark 2.4
  • Definition 2.5
  • Theorem 2.6
  • Remark 2.7
  • Proposition 2.8
  • Lemma 2.9
  • Example 2.10
  • ...and 48 more